where \(o\)  denotes the origin of \({\mathbb {R}}^d\) . The second equality follows from the fact that almost surely \(X_1,\dots ,X_N\)  are in general position and span \({\mathbb {R}}^d\)  linearly. We call the numbers \(P_{d,N}\)  the Wendel probabilities, since they were first explicitly determined by Wendel [1]} (the proof is reproduced in [2]}). The result is that
\(P_{d,N}= \frac{C(N,d)}{2^N},\qquad C(N,d) := 2\sum _{i=0}^{d-1}\binom{N-1}{i}.\) 
